Math · College algebra · Worked example

Factor out a horizontal scale before shifting

Describe the graph of g(x) = √(2x − 6) compared with f(x) = √x.

g (x) = \sqrt{2 x - 6}

Factor the input

Pull the 2 out of the whole input so that the shift appears on its own.

\sqrt{2 x - 6} = \sqrt{2 (x - 3)}

Read b and h

b = 2 halves horizontal distances, and h = 3 shifts right 3. There is no vertical change.

Move key points

Each point (x, y) on √x moves to (x/2 + 3, y). The starting point (0, 0) moves to (3, 0), (1, 1) to (3.5, 1) and (4, 2) to (5, 2). The domain becomes x ≥ 3.

(4, 2) \to (\frac{4}{2} + 3, 2) = (5, 2)

Check with the formula

The moved points satisfy the formula for g.

\begin{array}{l} g (x) = \sqrt{2 x - 6} \\ g (3) = 0 \\ g (3.5) = 1 \\ g (5) = 2 \end{array}

Result

Squeeze horizontally by a factor of 2, then shift right 3: the graph starts at (3, 0) and passes through (5, 2).

Your turn

Where is the vertex of y = (3x + 6)²?

Show the answer and explanation

(−2, 0).

Factor the input: 3x + 6 = 3(x + 2), so the graph of x² is squeezed horizontally by 3 and shifted left 2. The vertex moves from (0, 0) to (−2, 0). Expanding the square gives the same function as 9(x + 2)².

\begin{array}{l} y = {(3 x + 6)}^{2} \\ y = 9 {(x + 2)}^{2} \end{array}

Keep exploring

In Graph, compare √(2x − 6) with √(2x) − 6. The second is squeezed but shifted down 6, not right: where the constant sits decides the direction.

Return to the concept →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

OpenStax College Algebra 2e — Transformation of functions

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